Secondly, we apply the dispersive sum rules [1]}, [2]} to
amplitudes with various helicity configurations of the external gravitons.Flat space
superconvergence considered in Ref. [3]} is a
particular example of these more general sum rules. We derive various bounds on the inelastic scattering (the one in which the final and initial state gravitons have different helicities) in terms of the elastic one (see e.g. Ref. [4]}). We also place a
precise bound on the \(R^3\)  coefficient in terms of the \(R^4\) 
coefficient (see (REF )). Such a bound translates the problem
of making the analysis of Ref. [5]} quantitatively
precise to the problem of the bounding the leading \(R^4\)  contact
coefficient in terms of the gap of the theory. This has been recently
done in Ref. [6]} in a similar perturbative setting
for \(D=10\)  maximal supergravity; see also Ref. [7]}
for the nonperturbative analysis of the same problem. It would, of course, be very
interesting to generalize these studies to more general cases of
graviton scattering.
